S-integrality for families of ordinary K3 surfaces and algebraicity theorems

Abstract: We prove an $S$-integrality theorem for special divisors on GSpin Shimura varieties in positive characteristic. Let $C$ be a generically ordinary curve in such a Shimura variety not contained in any special divisor. Then, for any increasing sequence of prime-to-$p$ positive integers $m_i$ and any finite set of closed points $S\subset C$, we prove that $C\setminus S$ meets the special divisor $Z(m_i)$ for all but finitely many $i$. The key new input is an algebraicity theorem for formal special endomorphisms which allows us to use techniques from Diophantine approximation. We prove this algebraicity theorem using a punctual monodromy theorem and a positive-characteristic analogue of the Mumford--Tate conjecture.